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\title{\mykant: Implementing Observational Equality}
\author{Francesco Mazzoli \texttt{<fm2209@ic.ac.uk>}}
\date{June 2013}

\begin{document}
\frame{\titlepage}

\begin{frame}
  \frametitle{Theorem provers are short-sighted}

  Two functions dear to the Haskell practitioner:
  \[
  \begin{array}{@{}l}
    \myfun{map} : (\myb{a} \myarr \myb{b}) \myarr \mylist{\myb{a}} \myarr \mylist{\myb{b}} \\
      \begin{array}{@{}l@{\myappsp}c@{\myappsp}c@{\ }c@{\ }l}
        \myfun{map} & \myb{f} & \mynil & = & \mynil \\
        \myfun{map} & \myb{f} & (\myb{x} \mycons \myb{xs}) & = & \myapp{\myb{f}}{\myb{x}} \mycons \myfun{map} \myappsp \myb{f} \myappsp \myb{xs} \\
      \end{array}
      \\
      \ \\
    (\myfun{${\circ}$}) : (\myb{b} \myarr \myb{c}) \myarr (\myb{a} \myarr \myb{b}) \myarr (\myb{a} \myarr \myb{c}) \\
    (\myb{f} \mathbin{\myfun{$\circ$}} \myb{g}) \myappsp \myb{x} = \myapp{\myb{g}}{(\myapp{\myb{f}}{\myb{x}})}
  \end{array}
  \]
\end{frame}

\begin{frame}
  \frametitle{Theorem provers are short-sighted}
  $\myfun{map}$'s composition law states that:
  \[
    \forall \myb{f} {:} (\myb{b} \myarr \myb{c}), \myb{g} {:} (\myb{a} \myarr \myb{b}). \myfun{map}\myappsp \myb{f} \mycomp \myfun{map}\myappsp \myb{g} \myeq \myfun{map}\myappsp (\myb{f} \mycomp \myb{g})
  \]
  We can convince Coq or Agda that
  \[
    \forall \myb{f} {:} (\myb{b} \myarr \myb{c}), \myb{g} {:} (\myb{a} \myarr \myb{b}), \myb{l} {:} \mylist{\myb{a}}. (\myfun{map}\myappsp \myb{f} \mycomp \myfun{map}\myappsp \myb{g}) \myappsp \myb{l} \myeq \myfun{map}\myappsp (\myb{f} \mycomp \myb{g}) \myappsp \myb{l}    
  \]
  But we cannot get rid of the $\myb{l}$.  Why?
\end{frame}

\begin{frame}
  \frametitle{Observational equality and \mykant}

  \emph{Observational equality} is a solution to this and other equality
  annoyances.

  \mykant\ is a system making observational equality more usable.

  The theory of \mykant\ is complete, the practice, not quite.
\end{frame}

\begin{frame}
  \frametitle{Theorem provers, dependent types} First class types: we
  can return them, have them as arguments, etc.
  \[
  \begin{array}{@{}l@{\ }l@{\ \ \ }l}
    \mysyn{data}\ \myempty & & \text{No members.} \\
    \mysyn{data}\ \myunit  & = \mytt & \text{One member.} \\
    \mysyn{data}\ \mynat   & = \mydc{zero} \mydcsep \mydc{suc}\myappsp\mynat & \text{Natural numbers.}
  \end{array}
  \]
  $\myempty : \mytyp$, $\myunit : \mytyp$, $\mynat : \mytyp$.

  $\myunit$ is trivially inhabitable: it corresponds to $\top$ in
  logic.

  $\myempty$ is \emph{not} inhabitable: it corresponds to $\bot$.

\end{frame}

\begin{frame}
  \frametitle{Theorem provers, dependent types}

  We can express relations:
  \[
  \begin{array}{@{}l}
    (\myfun{${>}$}) : \mynat \myarr \mynat \myarr \mytyp \\
    \begin{array}{@{}c@{\,}c@{\,}c@{\ }l}
      \mydc{zero} & \mathrel{\myfun{$>$}} & \myb{m} & = \myempty \\
      \myb{n}     & \mathrel{\myfun{$>$}} & \mydc{zero} & = \myunit \\
      (\mydc{suc} \myappsp \myb{n}) & \mathrel{\myfun{$>$}} & (\mydc{suc} \myappsp \myb{m}) & = \myb{n} \mathrel{\myfun{$>$}} \myb{m}
    \end{array}
  \end{array}
  \]

  A term of type $\myb{m} \mathrel{\myfun{$>$}} \myb{n}$ represents a
  \emph{proof} that $\myb{n}$ is indeed greater than $\myb{n}$.
  \[
  \begin{array}{@{}l}
    3 \mathrel{\myfun{$>$}} 1 \myred \myunit \\
    2 \mathrel{\myfun{$>$}} 2 \myred \myempty \\
    0 \mathrel{\myfun{$>$}} \myb{m} \myred \myempty
  \end{array}
  \]

  Thus, proving that $2 \mathrel{\myfun{$>$}} 2$ corresponds to proving
  falsity, while $3 \mathrel{\myfun{$>$}} 1$ is fine.
\end{frame}

\begin{frame}
  \frametitle{Example: safe $\myfun{head}$ function}

  \[
  \begin{array}{@{}l}
    \mysyn{data}\ \mylistt{\myb{A}} = \mynil \mydcsep \myb{A} \mycons \mylistt{\myb{A}} \\
    \ \\
    \myfun{length} : \mylistt{\myb{A}} \myarr \mynat \\
    \begin{array}{@{}l@{\myappsp}c@{\ }c@{\ }l}
      \myfun{length} & \mynil & = & \mydc{zero} \\
      \myfun{length} & (\myb{x} \mycons \myb{xs}) & = & \mydc{suc} \myappsp (\myfun{length} \myappsp \myb{xs})
    \end{array} \\
    \ \\
    \myfun{head} : \myfora{\myb{l}}{\mytyc{List}\myappsp\myb{A}}{ \myfun{length}\myappsp\myb{l} \mathrel{\myfun{$>$}} 0 \myarr \myb{A}} \\
    \begin{array}{@{}l@{\myappsp}c@{\myappsp}c@{\ }c@{\ }l}
      \myfun{head} & \mynil & \myb{p} & = & \myhole{?} \\
      \myfun{head} & (\myb{x} \mycons \myb{xs}) & \myb{p} & = & \myb{x}
    \end{array}
  \end{array}
  \]

  The type of $\myb{p}$ in the $\myhole{?}$ is $\myempty$, since
  \[\myfun{length} \myappsp \mynil \mathrel{\myfun{$>$}} 0 \myred 0 \mathrel{\myfun{$>$}} 0 \myred \myempty \]
\end{frame}


\begin{frame}
  \frametitle{Example: safe $\myfun{head}$ function}

  \[
  \begin{array}{@{}l}
    \mysyn{data}\ \mylistt{\myb{A}} = \mynil \mydcsep \myb{A} \mycons \mylistt{\myb{A}} \\
    \ \\
    \myfun{length} : \mytyc{List}\myappsp\myb{A} \myarr \mynat \\
    \begin{array}{@{}l@{\myappsp}c@{\ }c@{\ }l}
      \myfun{length} & \mynil & = & \mydc{zero} \\
      \myfun{length} & (\myb{x} \mycons \myb{xs}) & = & \mydc{suc} \myappsp (\myfun{length} \myappsp \myb{xs})
    \end{array} \\
    \ \\
    \myfun{head} : \myfora{\myb{l}}{\mytyc{List}\myappsp\myb{A}}{ \myfun{length}\myappsp\myb{l} \mathrel{\myfun{$>$}} 0 \myarr \myb{A}} \\
    \begin{array}{@{}l@{\myappsp}c@{\myappsp}c@{\ }c@{\ }l}
      \myfun{head} & \mynil & \myb{p} & = & \myabsurd \myappsp \myb{p} \\
      \myfun{head} & (\myb{x} \mycons \myb{xs}) & \myb{p} & = & \myb{x}
    \end{array}
  \end{array}
  \]

  Where $\myfun{absurd}$ corresponds to the logical \emph{ex falso
    quodlibet}---given $\myempty$, we can get anything:
  \[
  \myfun{absurd} : \myempty \myarr \myb{A}
  \]
\end{frame}

\begin{frame}
  \frametitle{How do we type check this thing?}
  \[
  \myfun{head} \myappsp \mylistt{3} : \myfun{length} \myappsp \mylistt{3} \mathrel{\myfun{$>$}} 0 \myarr \mynat
  \]

  Will $\mytt : \myunit$ do as an argument?  In other words, when type
  checking, do we have that
  \[
  \begin{array}{@{}c@{\ }c@{\ }c}
    \myunit & \mydefeq & \myfun{length} \myappsp \mylistt{3} \mathrel{\myfun{$>$}} 0 \\
    \myfun{length} \myappsp \mynil \mathrel{\myfun{$>$}} 0 & \mydefeq & \myempty \\
    (\myabs{\myb{x}\, \myb{y}}{\myb{y}}) \myappsp \myunit \myappsp \myappsp \mynat & \mydefeq & (\myabs{\myb{x}\, \myb{y}}{\myb{x}}) \myappsp \mynat \myappsp \myunit \\
    & \vdots &
  \end{array}
  \]
  ?
\end{frame}

\begin{frame}
  \frametitle{Definitional equality}
  
  The type checker needs a notion of equality between types.

  We reduce terms `as far as possible' (to their \emph{normal form}) and
  then compare them syntactically:
  \[
  \begin{array}{@{}r@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }l}
    \myunit & \myredd & \myunit & \mydefeq & \myunit & \myreddd & \myfun{length} \myappsp \mylistt{3} \mathrel{\myfun{$>$}} 0 \\
    \myfun{length} \myappsp \mynil \mathrel{\myfun{$>$}} 0 & \myredd & \myempty & \mydefeq & \myempty & \myreddd & \myempty \\
    (\myabs{\myb{x}\, \myb{y}}{\myb{y}}) \myappsp \myunit \myappsp \myappsp \mynat & \myredd & \mynat & \mydefeq & \mynat & \myreddd & (\myabs{\myb{x}\, \myb{y}}{\myb{x}}) \myappsp \mynat \myappsp \myunit \\
    & & & \vdots & & & 
  \end{array}
  \]

  This equality, $\mydefeq$, takes the name of \emph{definitional} equality.
\end{frame}

\begin{frame}
  \frametitle{Propositional equality} Using definitional equality, we
  can give the user a type-level notion of term equality.
  \[
  (\myeq) : \myb{A} \myarr \myb{A} \myarr \mytyp
  \]
  We introduce members of $\myeq$ by reflexivity:
  \[
  \myrefl\myappsp\mytmt : \mytmt \myeq \mytmt
  \]
  So that $\myrefl$ will relate definitionally equal terms:
  \[
  \myrefl\myappsp 5 : (3 + 2) \myeq (1 + 4)\ \text{since}\ (3 + 2) \myeq (1 + 4) \myredd 5 \myeq 5
  \]
  Then we can use a substitution law to derive other
  laws---transitivity, congruence, etc.
\end{frame}

\begin{frame}
\frametitle{The problem with prop. equality}
Going back to $\myfun{map}$, we can prove that
\[    \forall \myb{f} {:} (\myb{b} \myarr \myb{c}), \myb{g} {:} (\myb{a} \myarr \myb{b}), \myb{l} {:} \mylist{\myb{a}}. (\myfun{map}\myappsp \myb{f} \mycomp \myfun{map}\myappsp \myb{g}) \myappsp \myb{l} \myeq \myfun{map}\myappsp (\myb{f} \mycomp \myb{g}) \myappsp \myb{l}    \]
Because we can prove, by induction on $\myb{l}$, that we will always get definitionally equal lists.

But without the $\myb{l}$, we cannot compute, so we are stuck with
\[
\myfun{map}\myappsp \myb{f} \mycomp \myfun{map}\myappsp \myb{g} \not\mydefeq \myfun{map}\myappsp (\myb{f} \mycomp \myb{g})
\]
\end{frame}

\begin{frame}
  \frametitle{The solution}

  \emph{Observational} equality, instead of basing its equality on
  definitional equality, looks at the structure of the type to decide:
  \[
  \begin{array}{@{}l}
    (\myfun{map}\myappsp \myb{f} \mycomp \myfun{map}\myappsp \myb{g} : \mylistt{\myb{A_1}} \myarr \mylistt{\myb{C_1}}) \myeq (\myfun{map}\myappsp (\myb{f} \mycomp \myb{g}) : \mylistt{\myb{A_2}} \myarr \mylistt{\myb{C_2}}) \myred \\
    \myind{2} (\myb{l_1} : \myb{A_1}) \myarr (\myb{l_2} : \myb{A_2}) \myarr (\myb{l_1} : \myb{A_1}) \myeq (\myb{l_2} : \myb{A_2}) \myarr \\
    \myind{2} ((\myfun{map}\myappsp \myb{f} \mycomp \myfun{map}\myappsp \myb{g}) \myappsp \myb{l} : \mylistt{\myb{C_1}}) \myeq (\myfun{map}\myappsp (\myb{f} \mycomp \myb{g}) \myappsp \myb{l} : \mylistt{\myb{C_2}})
  \end{array}
  \]
  This extends to other structures (tuples, inductive types, \dots).
  Moreover, if we can deem two \emph{types} equal, we can \emph{coerce}
  values from one to the other.
\end{frame}

\begin{frame}
  \frametitle{\mykant}

  Observational equality was described in a very restricted theory.

  \mykant\ aims to incorporate it in a more `practical' environment,
  where we have:
  \begin{itemize}
    \item User defined data types (inductive data and records).
    \item A type hierarchy.
    \item Partial type inference (bidirectional type checking).
    \item Type holes.
  \end{itemize}
\end{frame}

\begin{frame}
  \frametitle{Inductive data}
  Good old Haskell data types:
  \[
  \mysyn{data}\ \mytyc{List}\myappsp \myb{A} = \mynil \mydcsep \myb{A} \mycons \mytyc{List}\myappsp\myb{A}
  \]
  But instead of general recursion and pattern matching, we have
  structural induction:
  \[
  \begin{array}{@{}l@{\ }l}
    \mytyc{List}.\myfun{elim} : & (\myb{P} : \mytyc{List}\myappsp\myb{A} \myarr \mytyp) \myarr \\
    & \myb{P} \myappsp \mynil \myarr \\
    & ((\myb{x} : \myb{A}) \myarr (\myb{l} : \mytyc{List}\myappsp \myb{A}) \myarr \myb{P} \myappsp \myb{l} \myarr \myb{P} \myappsp (\myb{x} \mycons \myb{l})) \myarr \\
    & (\myb{l} : \mytyc{List}\myappsp\myb{A}) \myarr \myb{P} \myappsp \myb{l}
  \end{array}
  \]
  Reduction:
  \[
  \begin{array}{@{}l@{\ }l}
    \mytyc{List}.\myfun{elim} \myappsp \myse{P} \myappsp \myse{pn} \myappsp \myse{pc} \myappsp \mynil & \myred \myse{pn} \\
    \mytyc{List}.\myfun{elim} \myappsp \myse{P} \myappsp \myse{pn} \myappsp \myse{pc} \myappsp (\mytmm \mycons \mytmn) & \myred \myse{pc} \myappsp \mytmm \myappsp \mytmn \myappsp (\mytyc{List}.\myfun{elim} \myappsp \myse{P} \myappsp \myse{pn} \myappsp \myse{ps} \myappsp \mytmt )
  \end{array}
  \]
\end{frame}

\begin{frame}
  \frametitle{Records}
  But also records:
  \[
  \mysyn{record}\ \mytyc{Tuple}\myappsp\myb{A}\myappsp\myb{B} = \mydc{tuple}\ \{ \myfun{fst} : \myb{A}, \myfun{snd} : \myb{B} \}
  \]
  Where each field defines a projection from instances of the record:
  \[
  \begin{array}{@{}l@{\ }c@{\ }l}
    \myfun{fst} & : & \mytyc{Tuple}\myappsp\myb{A}\myappsp\myb{B} \myarr \myb{A} \\
    \myfun{snd} & : & \mytyc{Tuple}\myappsp\myb{A}\myappsp\myb{B} \myarr \myb{B}
  \end{array}
  \]
  Where the projection's reduction rules are predictably
  \[
  \begin{array}{@{}l@{\ }l}
    \myfun{fst}\myappsp&(\mydc{tuple}\myappsp\mytmm\myappsp\mytmn) \myred \mytmm \\
    \myfun{snd}\myappsp&(\mydc{tuple}\myappsp\mytmm\myappsp\mytmn) \myred \mytmn \\
  \end{array}
  \]
\end{frame}

\begin{frame}
  \frametitle{Dependend defined types} \emph{Unlike} Haskell, we can
  have fields of a data constructor to depend on earlier fields:
  \[
  \begin{array}{@{}l}
    \mysyn{record}\ \mytyc{Tuple}\myappsp(\myb{A} : \mytyp)\myappsp(\myb{B} : \myb{A} \myarr \mytyp) = \\
    \myind{2}\mydc{tuple}\ \{ \myfun{fst} : \myb{A}, \myfun{snd} : \myb{B}\myappsp\myb{fst} \}
  \end{array}
  \]
  $\mytyc{Tuple}$ takes a $\mytyp$, $\myb{A}$, and a predicate from
  elements of $\myb{A}$ to types, $\myb{B}$.

  This way, the \emph{type} of the second element depends on the
  \emph{value} of the first:
  \[
  \begin{array}{@{}l@{\ }l}
    \myfun{fst} & : \mytyc{Tuple}\myappsp\myb{A}\myappsp\myb{B} \myarr \myb{A} \\
    \myfun{snd} & : (\myb{x} : \mytyc{Tuple}\myappsp\myb{A}\myappsp\myb{B}) \myarr \myb{B} \myappsp (\myfun{fst} \myappsp \myb{x})
  \end{array}
  \]
\end{frame}

\begin{frame}
  \frametitle{Type hierarchy}
  Up to now, we have thrown $\mytyp$ around, as `the type of types'.

  But what is the type of $\mytyp$ itself?  The simple way out is
  \[
  \mytyp : \mytyp
  \]
  This solution is not only simple, but inconsistent, for the same
  reason that the notion of a `powerset' in na{\"i}ve set theory is.

  Instead, following Russell, we shall have
  \[
  \{\mynat, \mybool, \mytyc{List}\myappsp\mynat, \cdots\} : \mytyp_0 : \mytyp_1 : \cdots
  \]
  We talk of types in $\mytyp_0$ as `smaller' than types in $\mytyp_1$.
\end{frame}

\begin{frame}
  \frametitle{Cumulativity and typical ambiguity}
  Is it OK to take
  \[ \mytyp_0 : \mytyp_2 \]
  ?
\end{frame}

\begin{frame}
\begin{center}
{\Huge Questions?}
\end{center}
\end{frame}

\end{document}